1. Field of the Invention
The present invention relates generally to the fields of neural networks. More particularly, it concerns using neural networks for backlash compensation in mechanical systems.
2. Description of Related Art
A general class of industrial motion control systems has the structure of a dynamical system, usually of the Lagrangian form, preceded by some nonlinearities in the actuator, which may be deadzone, backlash, saturation, or the like. This includes xy-positioning tables, robot manipulators, robots, overhead crane mechanisms, machine tools, vehicle suspension systems, and many more mechanical systems. Problems associated with nonlinearities may be exacerbated when the required accuracy is high, as in micropositioning devices. Due to the nonanalytic nature of actuator nonlinearities and the fact that their exact nonlinear functions are unknown, such systems present a serious challenge for the control design engineer. In particular, inverting backlash nonlinearity even for known backlash models is extremely difficult since the nonlinearity appears in the feedforward path. It would therefore be advantageous to have the ability to both estimate the nonlinearities and compensate: for their effects in a mechanical system. It would be especially advantageous to have the ability to estimate and compensate for the effects of backlash in a mechanical system.
Neural Networks (NN) have been used extensively in feedback control systems. Specifically, it has been observed that the use of Proportional-Derivative (PD) controllers may result in limit cycles if actuators have deadzones or backlash. Rigorous results for motion tracking of such systems are notably sparse, though ad hoc techniques relying on simulations for verification of effectiveness are prolific.
Although some NN applications show at least a degree of utility, most applications, unfortunately, are ad hoc with no demonstration of stability. The stability proofs that do exist rely almost invariably on the universal approximation property for NN. However, in most real industrial control systems there are nonsmooth functions (piecewise continuous) for which approximation results in the literature are sparse or nonexistent. Examples of phenomena involving nonsmooth functions include, but are not limited to, deadzone, friction, and backlash. Though there do exist some results for piecewise continuous functions, traditional attempts to approximate jump functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. It would therefore be advantageous to have the ability to estimate and compensate nonlinearities, including backlash nonlinearities, involving piecewise continuous functions with guaranteed close-loop stability.
Certain problems facing the field enumerated above are not intended to be exhaustive but rather are among many which tend to impair the effectiveness of previously known estimation and compensation schemes. Other noteworthy problems may and do exist; however, those presented above are sufficient to demonstrate that methods of compensation, including backlash compensation, appearing in the art have not been altogether satisfactory.
The present invention relates to a dynamic inversion compensation scheme for backlash but may be applied for compensation of a large class of invertible dynamical nonlinearities. A general model of backlash is used, which is not required to be symmetric. The compensator of the present invention uses a backstepping technique with neural networks (NN) for inverting backlash nonlinearity in the feedforward path. Compared with adaptive backstepping control schemes, the present disclosure does not require unknown parameters to be linear parametrizable, and no regression matrices are needed. Instead of a derivative, the present invention uses a filtered derivative.
In the detailed disclosure found below, full rigorous stability proofs are given using the filtered derivatives of the present invention. In accordance with the present disclosure, small tracking errors are guaranteed, and bounded NN weights are presented. Also shown below is a detailed analysis using a real filter needed to calculate the derivative of signals used in inverse dynamics design.
The NN of the present invention may be used for compensating dynamic inversion error. A modified Hebbian tuning algorithm for the NN is designed in order to simplify the computational burden in multi-axes mechanical systems with backlash. Although the nonlinear system described herein is assumed to be in Brunovsky form, those having skill in the art will recognize that techniques of the present invention may be applied to different forms as well.
As shown in the Examples section of this disclosure, simulation results show that a NN backlash compensator according to the present invention may significantly reduce the degrading effect of backlash nonlinearity.
The present invention provides for several advantages and offers several features that differentiate it from previously-known methodology. Some of these advantages and features include, but are not limited to, the following:
(1) Most backlash compensators using older control technology do not have any performance or stability guarantees. This makes their acceptance by industry questionable;
(2) Other modern backlash compensators that do not use neural nets, e.g. those based on adaptive control, require a strong xe2x80x9clinear in the parameters assumptionxe2x80x9d that does not often hold for actual industrial systems. This restricts the types of backlashes that can be canceled;
(3) The inventors are not aware of any backlash compensation method based on neural networks. Although there are certain, compensation methods for other actuator nonlinearities (deadzone and friction) based on neural nets, they do not provide design algorithms based on mathematical proofs that guarantee stability of the controlled system. Instead, they use standard xe2x80x98backpropagationxe2x80x99 weight tuning, which cannot provide stability in all situations;
(4) Other backlash compensation techniques based on fuzzy logic use an ad hoc structure that does not have a rigorous stability analysis;
(5) Other dynamic inversion approaches require a non-implementable ideal differentiator. The present invention uses a filter derivative, which is easy to implement; and
(6) Other dynamic inversion approaches based on neural networks use more complex tuning algorithms. Hebbian tuning, disclosed herein, is computationally simper and more suitable for implementation on actual industrial devices.
Noteworthy features of the present invention include, but are not limited to, the following:
(1) The invention is a xe2x80x98model-freexe2x80x99 approach. Specifically, a neural net may be used for backlash compensation so that a mathematical model of the backlash is not needed;
(2) The invention has application for a large class of xe2x80x98dynamicxe2x80x99 actuator nonlinearities, not only backlash;
(3) The neural net design algorithm based on dynamic inversion with rigorous mathematical stability proofs of the present invention shows how to guarantee closed-loop stability and performance;
(4) The modified Hebbian tuning algorithm presented herein offers computational advantages over gradient descent based algorithms. No preliminary off-line tuning is needed. Weights are tuned on-line, and the algorithms guarantee stability of the controlled system;
(5) Neural net weight initialization according to the present invention is simple using algorithms disclosed herein. Weights may be initialized so that the neural net output is zero, and a proportional-derivative (PD) control loop keeps the system stable initially until the weights begin to learn; and
(6) A filtered derivative is used in the feedforward loop of this invention, instead of an ideal derivative, which cannot be implemented as in other dynamic inversion approaches.
Potential markets and applications for the present invention are vast and include, but are not limited to, electrical components fabrication, the auto industry, military applications, robotics, mechanical processes, and biomedical engineering.
In one respect, the invention is an adaptive neural network compensator for compensating backlash of a mechanical system. As used herein, by xe2x80x9cmechanical systemxe2x80x9d it is meant any system exhibiting a nonlinearity. Examples of mechanical systems include, but are not limited to, actuators, robots, x-y manipulators, cranes, and any system showing any nonlinearity in response. The compensator includes a feedforward path, a proportional derivative tracking loop, a filter, and a neural network. The proportional derivative tracking loop includes a proportional derivative path in the feedforward path. The filter is in the feedforward path and is configured to take a filtered derivative of a tracking error signal of the compensator to determine a filtered tracking error signal. The neural network is in the feedforward path and is coupled to the filter. The neural network is configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system.
In other respects, the neural network may be tuned according to the following equations:
{circumflex over ({dot over (V)})}=xe2x88x92T∥{tilde over (xcfx84)}∥xnn{circumflex over ("sgr")}Txe2x88x92kT∥{tilde over (xcfx84)}∥{circumflex over (V)}; and
{circumflex over ({dot over (W)})}=xe2x88x92S{circumflex over ("sgr")}{tilde over (xcfx84)}Txe2x88x92kS∥{tilde over (xcfx84)}∥Ŵ,
where V-hat represents an estimate of first layer neural network weights, W-hat represents an estimate of second layer neural network weights, Sigma-hat represents an estimate of neural network hidden-layer output, Tau-tilde represents error in an estimate of an input to the mechanical system, xnn represents an input vector to the neural network, S and T are design weighting matrices, and k is a design convergence parameter. The filtered tracking error signal is uniformly ultimately bounded. Stability proofs disclosed herein guarantee the bounded nature of the filtered tracking error signal. The weight estimates of the neural network may be uniformly ultimately bounded. Stability proofs disclosed herein guarantee the bounded nature of the weight estimates. The filtered tracking error signal and weight estimates of the neural network may be uniformly ultimately bounded. Stability proofs disclosed herein guarantee the bounded nature of the filtered tracking error signal and weight estimates together. The proportional derivative tracking loop may be configured to provide stable feedback control of the mechanical system while weights of the neural network are adjusted from initialization values. The mechanical system comprises an actuator or robot.
In another respect, the invention is an adaptive neural network compensator for compensating backlash of a mechanical system, including a filter, a neural network, and means for tuning the neural network. The filter is in a feedforward path and is configured to determine a filtered tracking error signal. The neural network is in the feedforward path and is configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system. The means for tuning the neural network ensures that the filtered tracking error signal of the compensator is uniformly ultimately bounded.
In another respect, the invention is an adaptive neural network compensator for compensating backlash of a mechanical system, including a feedforward path, a proportional derivative tracking loop, a neural network, and a nonlinear estimate feedback loop. The proportional derivation tracking loop includes a proportional derivative path in the feedforward path. The neural network is in the feedforward path. The nonlinear estimate feedback loop is coupled to the feedforward path.
In another respect, the invention is a method of adaptively compensating backlash in a mechanical system. An inverse of the backlash is estimated using a neural network in a feedforward path. A filtered derivative of a tracking error signal of the compensator is taken using a filter in the feedforward path to form a filtered tracking error signal. Weights of the neural network are adjusted as a function of the filtered tracking error signal using a Hebbian tuning algorithm to achieve closed loop stability. The inverse is applied to an input of the mechanical system to compensate the backlash.
In other respects, initial weights V of the neural network may be selected randomly and initial weights W of the neural network may be set to zero. The method may also include providing stable feedback control of the mechanical system while weights of the neural network are adjusted from initialization values with a proportional derivative tracking loop comprising a proportional derivative path in the feedforward path.